Why is Institutional holding considered a proxy for short-sale costs? In other words, Why are firms with lower institutional ownership more likely to be a subject of short-sale constraints?

For example Asquith, Pathak, and Ritter (2005) use institutional ownership as a proxy for the lendable supply of shares and define short sale-constrained stocks as having both high RSI (relative short interest) and low institutional ownership.

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Assuming it is true, most likely due to market/institutional constraints.

Institutional investors hold their assets at a custodial bank. Those banks operate securities lending programs which lend out shares for short sales to make a small incremental additional revenue. On top of that, institutional investors have - at least in the past - been huge holders of index funds (trillions of dollars). Index funds seldom need to 'recall' lent shares because a portfolio manager (as happens in an active fund) sours on a drooping stock and wants to sell it - precisely when the borrower, who is short, is profiting. ETFs are large enough that this, by the way, may have changed in recent years.

Retail investors don't participate in lending as much. Or, at least, they often don't know; the brokerage firms sometimes have an "opt-out" buried deep in the account docs. But, at the same time, retail investors are an unsteady supply of lendable stocks since they are (probably) more likely to sell when a stock that has been lent starts dropping fast.

BTW, when retail brokerage does this it is called "lending against the box", the box being, euphemistically, the big supply of securities in client accounts: they might borrow from one client to lend stock to another to short-sell. If you have stock holdings, you should try to ask your broker to cut you in on the lending revenue from your shares or stop lending them. You might possibly get some revenue.